A Novel Intelligent Method for the State of Charge Estimation of Lithium-Ion Batteries Using a Discrete Wavelet Transform-Based Wavelet Neural Network

State of charge (SOC) estimation is becoming increasingly important, along with electric vehicle (EV) rapid development, while SOC is one of the most significant parameters for the battery management system, indicating remaining energy and ensuring the safety and reliability of EV. In this paper, a hybrid wavelet neural network (WNN) model combining the discrete wavelet transform (DWT) method and adaptive WNN is proposed to estimate the SOC of lithium-ion batteries. The WNN model is trained by Levenberg-Marquardt (L-M) algorithm, whose inputs are processed by discrete wavelet decomposition and reconstitution. Compared with back-propagation neural network (BPNN), L-M based BPNN (LMBPNN), L-M based WNN (LMWNN), DWT with L-M based BPNN (DWTLMBPNN) and extend Kalman filter (EKF), the proposed intelligent SOC estimation method is validated and proved to be effective. Under the New European Driving Cycle (NEDC), the mean absolute error and maximum error can be reduced to 0.59% and 3.13%, respectively. The characteristics of high accuracy and strong robustness of the proposed method are verified by comparison study and robustness evaluation results (e.g., measurement noise test and untrained driving cycle test).


Introduction
The battery management system (BMS) is responsible for monitoring power batteries' complete information to guarantee the electric vehicle (EV) performance, because the batteries' parameters such as current, voltage, resistance, and temperature are of importance, indicating the safety and normality of the power system [1].State of Charge (SOC) has an undoubted critical position for BMS to realize management functions, and it indicates the remaining energy, whose accuracy is of great significance for service safety and life of batteries [2,3].Nevertheless, the limitation on the progress of BMS is mostly due to SOC unmeasurable and dynamic properties that are similar to the characteristics of the batteries, which are influenced by various factors [4], such as discharge rate, ambient temperature, battery degeneration, and external disturbance.Therefore, the study of high accuracy SOC estimation methods is vitally important using measureable variables, such as current, voltage and temperature.
With the rapid development of EV and the increasing importance of BMS, numbers of estimation approaches have been proposed to monitor the SOC.The ampere-hour (A•h) integral or Coulomb counting method [5,6] and open-circuit voltage method [7] are simple to implement, but non-model Energies 2018, 11, 995 3 of 18 make the neural network model easy to establish and apply to the estimation method.Little research studies the combination of the hybrid WNN connecting method and the adaptive WNN.
The field of WNN-based SOC estimation needs more study.Most researches mainly focus on the denoising ability of discrete wavelet transform (DWT) [40][41][42].The hybrid WNN [43] -based SOC estimation methods; adaptive WNN-based [32] in 2005; and momentum-optimized, adaptive WNN [31]-based in 2013 SOC estimation methods are discussed, which have good performance facing the high nonlinear battery system, but the training algorithm is based on the steepest descent method, and the study of the hybrid WNN using the wavelet multi-resolution decomposition method to optimized the adaptive WNN is quite limited.The combination is supposed to have high accuracy estimation ability on SOC.
To solve the above problems, this paper combines the adaptive WNN with the hybrid WNN and proposes a kind of novel, intelligent SOC estimation method that is expressed as the DWT and Levenberg-Marquardt (L-M) algorithm-based WNN (DWTLMWNN).The adaptive WNN is trained by L-M algorithm and, importantly, the input signals are from the data processed by discrete wavelet decomposition and reconstitution.The SOC, in discharge process under the New European Driving Cycle (NEDC), is estimated by BPNN, EKF, and the proposed WNN-based methods as a comparison.Estimation robustness is discussed by the untrained driving cycle test and measurement noise test.
The remaining part of this paper is organized as follows.Section 2 introduces the adaptive WNN model.The L-M algorithm and DWT optimizing process are presented in Section 3. The experimental validation of the proposed method and discussion are proposed in the Section 4. In the end, the paper is summarized in Section 5.

Estimation Model Based on WNN
The definition of SOC is commonly defined as the ratio of the remaining capacity to the nominal capacity and formulated as in which i(τ) and C n , respectively, denote the battery current and the nominal capacity of the battery.
The adaptive WNN has good performance for the reason that the parameters of wavelet templates can be adaptively generated [39].As shown in Figure 1, there are three layers (input layer, hidden layer, and output layer) that consist of K, L, and M nodes, respectively.The input data and output data are presented by x k and o m .The weights ω kl and ω lm , the wavelet translation parameter b l , and wavelet dilation parameter a l are the main parameters adjusted corresponding to the learning process.x l is defined as the output of the hidden layer, which is formulated as The field of WNN-based SOC estimation needs more study.Most researches mainly focus on the denoising ability of discrete wavelet transform (DWT) [40][41][42].The hybrid WNN [43] -based SOC estimation methods; adaptive WNN-based [32] in 2005; and momentum-optimized, adaptive WNN [31]-based in 2013 SOC estimation methods are discussed, which have good performance facing the high nonlinear battery system, but the training algorithm is based on the steepest descent method, and the study of the hybrid WNN using the wavelet multi-resolution decomposition method to optimized the adaptive WNN is quite limited.The combination is supposed to have high accuracy estimation ability on SOC.
To solve the above problems, this paper combines the adaptive WNN with the hybrid WNN and proposes a kind of novel, intelligent SOC estimation method that is expressed as the DWT and Levenberg-Marquardt (L-M) algorithm-based WNN (DWTLMWNN).The adaptive WNN is trained by L-M algorithm and, importantly, the input signals are from the data processed by discrete wavelet decomposition and reconstitution.The SOC, in discharge process under the New European Driving Cycle (NEDC), is estimated by BPNN, EKF, and the proposed WNN-based methods as a comparison.Estimation robustness is discussed by the untrained driving cycle test and measurement noise test.
The remaining part of this paper is organized as follows.Section 2 introduces the adaptive WNN model.The L-M algorithm and DWT optimizing process are presented in Section 3. The experimental validation of the proposed method and discussion are proposed in the Section 4. In the end, the paper is summarized in Section 5.

Estimation Model Based on WNN
The definition of SOC is commonly defined as the ratio of the remaining capacity to the nominal capacity and formulated as in which ( ) i  and n C , respectively, denote the battery current and the nominal capacity of the battery.
The adaptive WNN has good performance for the reason that the parameters of wavelet templates can be adaptively generated [39].As shown in Figure 1, there are three layers (input layer, hidden layer, and output layer) that consist of K, L, and M nodes, respectively.The input data and output data are presented by k x and m o .The weights kl  and ' lm  , the wavelet translation parameter l b , and wavelet dilation parameter l a are the main parameters adjusted corresponding to the learning process.
' l x is defined as the output of the hidden layer, which is formulated as  Energies 2018, 11, 995 4 of 18 in which ψ(x) denotes the generating functions, and in this paper ψ(x) is defined as Morlet wavelet function: The outputs of output layer's nodes are formulated as Firstly, the process of training WNN is demonstrated basing the steepest descent algorithm as a basis of following section.The output mean square error is defined as The weights and wavelet parameters can be adjusted as follows: b * l = b l + ∆b l (10) in which ∆ω kl , ∆ω lm , ∆a l , and ∆b l are calculated as in which we have Energies 2018, 11, 995 in which ψ (1) (x) denotes the first derivatives of ψ(x).Substitute Equations ( 15)-( 21) into Equations ( 11)-( 14): Furthermore, Equations ( 22)-( 25) can be rewritten as in which δ m and δ l are defined as equivalent errors: Modifying the WNN for the process of SOC estimation, the number of output nodes is set to 1.

L-M Algorithm
L-M algorithm is an outstanding optimization method, which combines the characteristic of Gauss-Newton's method and the steepest descent algorithm [44].On account of the considerable Energies 2018, 11, 995 6 of 18 performance in much research [45][46][47], it is chosen as the learning algorithm of the WNN.The total mean square error of P sets of training data is defined as L-M algorithm is formulated as in which the Jacobian matrix J(h) is and with h i (i = 1, 2, . . ., I) representing the i-th term of parameters that need to be updated.The partial derivatives in J(h) are calculated similar to the steepest descent algorithm, but one modification at the output layer should be conducted.The equivalent error δ m in Equation ( 30) should be replaced by In each iteration process, the L-M learning algorithm proceeds as follows: a.
Calculate equivalent errors of each layer using Equation (39) and c.
The parameter µ in Equation (33), which decides the result closer to Gauss-Newton algorithm or the steepest descent algorithm, is multiplied by factor β when V(h)increases in present iteration; then, turn back to step d.Otherwise, it is divided by β when V(h) is decreased in present iteration; then, turn to the next iteration process until the maximum iteration.In Section 4.3, µ = 0.01 is selected as the initial value with β = 10.

Discrete Wavelet Transform
For a nonstationary signal, wavelet analysis is more effective than the Fourier analysis on account of the good localization in time and frequency domains [48].This paper preprocesses the input data using DWT and inverse discrete wavelet transform (IDWT).In terms of MRA, the reconstitution data of the approximate and detail signals are regarded as the inputs of the WNN.
The DWT of an original time series signal x(t), according to the wavelet analysis theory proposed by Percival and Walden in 2000, is defined as below [49] with ) and where ψ(t) denotes the mother wavelet, ψ * indicates the complex conjugate of mother wavelet, and a 0 , b 0 are constants.Dilation parameter j and translation k(j, k ∈ R) are two scaling parameters that determine the oscillatory frequency and shifted position, respectively.In practice, a 0 and b 0 are usually set as 2 and 1, respectively: with MRA provides a fast implementation method of DWT, which has good approximations of low-frequency components and good resolution of the details at high frequencies.Based on low-and high-pass filters, an original signal x(t) can be expressed as the J-level DWT representation as Equation ( 47) [50]: where J(J ≤ N) is the number of decomposition levels with maximum decomposition level N and φ(t) denotes the scaling function.The original signal x(t) can be separated into lower resolution components that are described by approximate coefficients a j,k and detailed coefficients d j,k through a serious of decomposition processes based on high-and low-pass filters and down-sampling.The inverse process can well reconstruct the decomposed signal by IDWT, which uses up-sampling and synthesis filter bank.
The DWT-based decomposition and reconstruction processes are shown in Figure 2, which describes the DWT-WNN-based SOC estimation methods as well.The input signals x(n), such as U(n), I(n), are decomposed into approximation (A 1 − A 3 ) and detail (D 1 − D 3 ) components, and the reconstructed components

Data Acquisition
In this paper, based on the experiment directed by the systems analysis tool ADVISOR [51] for EV, NEDC, UDDS (Urban Dynamometer Driving Schedule), and UKBC (the United Kingdom Bus Cycle), three driving cycles are tested, which are shown in

Method Validation and Comparison Study
In this paper, in order to validate the performance of DWTLMWNN, other SOC estimation methods based on BPNN, L-M-based BPNN (LMBPNN), L-M based WNN (LMWNN), DWT-based LMBPNN (DWTLMBPNN), and EKF are used to make a comparative study.The estimation process of DWT-based WNN is as shown in Figure 1.In order to denoise and keep most useful information, based on a preliminary experiment, the input data of the WNN is set as the reconstructed components

Method Validation and Comparison Study
In this paper, in order to validate the performance of DWTLMWNN, other SOC estimation methods based on BPNN, L-M-based BPNN (LMBPNN), L-M based WNN (LMWNN), DWT-based LMBPNN (DWTLMBPNN), and EKF are used to make a comparative study.The estimation process of DWT-based WNN is as shown in Figure 1.In order to denoise and keep most useful information, based on a preliminary experiment, the input data of the WNN is set as the reconstructed components x(n) A 3 and x(n) D 1 of the current and voltage, respectively.The DWT and IDWT processes are conducted in three decomposition levels using the Daubechies basis of order 5 (dB5).The training reconstructed components are normalized into the range of [-1, 1], which are transformed using the following formula: x mid = x max +x min 2 x = 2(x−x mid ) x max −x min (48) in which x max and x min , respectively, denote the maximum value and minimum value of original input variables, and x and x , respectively, denote original input variable and normalized input variable.On account of random selection of the initial WNN parameters, the estimation result may possess small fluctuations in the same conditions.The results shown in this paper are the moderate and effective ones.The number of nodes in hidden layer is selected as 10.
As shown in Figure 5 and Table 1, BPNN, LMBPNN, LMWNN, DWTLMBPNN, DWTLMWNN, and EKF are validated under the NEDC.The ANN-based methods in the experiments have the same training data, number of hidden layer nodes, and working environment.The comparative EKF SOC estimation method is based on the second-order equivalent circuit model.The identified parameters and tuning parameters are shown in Table 2. To make the comparison more reasonable, the initial state is set as [1;0;0], while the initial SOC is supposed to be 100%.Thus, the covariance of observation noise Q k is selected to be not very small as a balance.
According to Figure 5b-d, the application of L-M algorithm and DWT method makes effective improvement on the SOC estimation accuracy.Based on the exact values in Table 1, it is found that L-M algorithm reduces the mean absolute error considerably and greatly improves the approximation ability according to the values of the correlation coefficient R but cannot reduce the maximum error to a desired level.DWT method can well reduce the maximum error of SOC estimation.Compared with model-based EKF estimation methods and BPNN-based estimation methods, WNN-based methods have the characteristics of low mean absolute error (e.g., only the mean absolute errors of LMWNN and DWTLMWNN below 1.00%), and the proposed method can solve the high maximum error problem (e.g., the maximum error is reduced more than 5% compared with LMWNN method).Furthermore, according to the results that DWT method reduces the mean absolute error and maximum error of LMBPNN by 0.03% and 1.74%, respectively, but of LMWNN by 0.23% and 5.23%, respectively, it is found that the estimation accuracy improvement using the combination of DWT method and adaptive WNN method is more considerable, which indicates the superiority of the proposed combination method.Figure 6 shows the error histogram of EKF, DWTLMBPNN, and DWTLMWNN, which directly supports that the DWTLMWNN SOC estimation method has advantage of estimation accuracy.Although the training time of ANN-based method is larger than the EKF based method unavoidably according to Table 1, the estimation time of ANN-based method is satisfying, which means the feasibility of actual SOC estimation application.combination of DWT method and adaptive WNN method is more considerable, which indicates the superiority of the proposed combination method.Figure 6 shows the error histogram of EKF, DWTLMBPNN, and DWTLMWNN, which directly supports that the DWTLMWNN SOC estimation method has advantage of estimation accuracy.Although the training time of ANN-based method is larger than the EKF based method unavoidably according to Table 1, the estimation time of ANNbased method is satisfying, which means the feasibility of actual SOC estimation application.combination of DWT method and adaptive WNN method is more considerable, which indicates the superiority of the proposed combination method.Figure 6 shows the error histogram of EKF, DWTLMBPNN, and DWTLMWNN, which directly supports that the DWTLMWNN SOC estimation method has advantage of estimation accuracy.Although the training time of ANN-based method is larger than the EKF based method unavoidably according to Table 1, the estimation time of ANNbased method is satisfying, which means the feasibility of actual SOC estimation application.

Robustness Evaluation
Influenced by electromagnetic interference or low precision sensors, the measured input data may not be that accurate.Therefore, noises are simulated as bias noises and random noises added to current and voltage to evaluate the robustness of proposed method.Moreover, different driving cycles have different charge and discharge forms.Therefore, tests on untrained driving cycle for the proposed method should be conducted.

Measurement Noise Test
As shown in Figure 7, the absolute values of positive and negative bias noises added to current and voltage are, respectively, 0.1 A and 0.01 V.The amplitudes of random noises are, respectively, 0.1 A and 0.01 V or 0.2 A and 0.02 V.According to Table 3, compared with EKF SOC estimation method, the DWTLMWNN SOC estimation method has great robustness against noises, especially against large random noises.Besides, the DWTLMWNN method is more stable than the EKF method for the reason that the mean absolute errors and maximum errors of different noise types for the DWTLMWNN method have much less variation, whose mean absolute errors range from 0.66% to 1.16% and maximum errors range from 3.62% to 5.12%, than for the EKF method, whose mean absolute errors range from 1.13% to 3.99% and maximum errors range from 3.32% to 11.44%.The results indicate that the DWTLMWNN method has a good performance against measurement noise and is more stable and accurate than the EKF SOC estimation method.

Robustness Evaluation
Influenced by electromagnetic interference or low precision sensors, the measured input data may not be that accurate.Therefore, noises are simulated as bias noises and random noises added to current and voltage to evaluate the robustness of proposed method.Moreover, different driving cycles have different charge and discharge forms.Therefore, tests on untrained driving cycle for the proposed method should be conducted.

Measurement Noise Test
As shown in Figure 7, the absolute values of positive and negative bias noises added to current and voltage are, respectively, 0.1 A and 0.01 V.The amplitudes of random noises are, respectively, 0.1 A and 0.01 V or 0.2 A and 0.02 V.According to Table 3, compared with EKF SOC estimation method, the DWTLMWNN SOC estimation method has great robustness against noises, especially against large random noises.Besides, the DWTLMWNN method is more stable than the EKF method for the reason that the mean absolute errors and maximum errors of different noise types for the DWTLMWNN method have much less variation, whose mean absolute errors range from 0.66% to 1.16% and maximum errors range from 3.62% to 5.12%, than for the EKF method, whose mean absolute errors range from 1.13% to 3.99% and maximum errors range from 3.32% to 11.44%.The results indicate that the DWTLMWNN method has a good performance against measurement noise and is more stable and accurate than the EKF SOC estimation method.In this section, the DWTLMWNN is trained by the data collected under the NEDC and UDDS, and then validated under the UKBC.As shown in Figure 8 and in Table 4, for the trained driving cycles and the untrained driving cycle, the DWTLMWNN SOC estimation method has good estimation performance in terms of the low mean absolute errors and maximum errors.Although the results show that the accuracy of SOC estimation under the NEDC reduces, the overall estimation results are ensured to a satisfying level.Compared with the EKF method, the DWTLMWNN SOC estimation method has a more reliable performance for the unexpected conditions.0.01 V/0.In this section, the DWTLMWNN is trained by the data collected under the NEDC and UDDS, and then validated under the UKBC.As shown in Figure 8 and in Table 4, for the trained driving cycles and the untrained driving cycle, the DWTLMWNN SOC estimation method has good estimation performance in terms of the low mean absolute errors and maximum errors.Although the results show that the accuracy of SOC estimation under the NEDC reduces, the overall estimation results are ensured to a satisfying level.Compared with the EKF method, the DWTLMWNN SOC estimation method has a more reliable performance for the unexpected conditions.

Conclusions
In this paper, DWTLMWNN, as a hybrid WNN model combining DWT method and adaptive WNN, is proposed to estimate the SOC of lithium-ion batteries.Comparing with BPNN, LMBPNN, LMWNN, DWTLMBPNN, and EKF, the proposed intelligent SOC estimation method is validated and proved to be more effective.The characteristics of high accuracy and strong robustness of the proposed method are verified by a comparison study and robustness evaluation results.
It is found that the proposed analyzed method can solve the high maximum error problem of the BPNN-or WNN-based methods (e.g., the maximum error of DWTLMWNN is reduced to 3.13% under the NEDC).Furthermore, the mean absolute error and maximum error are ensured in a satisfying level even if there is large measurement noise (e.g., 0.2 A/0.02 V random noises) or under untrained driving cycles.Therefore, the proposed method is suitable and of great significance for SOC estimation.

Conclusions
In this paper, DWTLMWNN, as a hybrid WNN model combining DWT method and adaptive WNN, is proposed to estimate the SOC of lithium-ion batteries.Comparing with BPNN, LMBPNN, LMWNN, DWTLMBPNN, and EKF, the proposed intelligent SOC estimation method is validated and proved to be more effective.The characteristics of high accuracy and strong robustness of the proposed method are verified by a comparison study and robustness evaluation results.
It is found that the proposed analyzed method can solve the high maximum error problem of the BPNN-or WNN-based methods (e.g., the maximum error of DWTLMWNN is reduced to 3.13% under the NEDC).Furthermore, the mean absolute error and maximum error are ensured in a satisfying level even if there is large measurement noise (e.g., 0.2 A/0.02 V random noises) or under untrained driving cycles.Therefore, the proposed method is suitable and of great significance for SOC estimation.
Future work may focus on verifying the proposed method using a high-rate load condition, and the robustness test on packet loss will be considered.Other learning algorithms of neural networks such as the Bayesian regularization method and scaled conjugate gradient will also be employed tentatively in the future.Additionally, the influence of temperature on the proposed method will be studied.

Figure 1 .
Figure 1.Structure of a three-layer WNN.

Figure 1 .
Figure 1.Structure of a three-layer WNN.
and x(n) D 3 are alternative as the inputs of WNN.
sampling and synthesis filter bank.The DWT-based decomposition and reconstruction processes are shown in Figure2, which describes the DWT-WNN-based SOC estimation methods as well.The input signals( )x n , such as ( ), ( ) U n I n , are decomposed into approximation 1 the inputs of WNN.

Figure 2 .
Figure 2. Structure of a three-level, DWT-based decomposition and reconstruction processes and DWT-WNN-based SOC estimation process.

Figure 2 .
Figure 2. Structure of a three-level, DWT-based decomposition and reconstruction processes and DWT-WNN-based SOC estimation process.

Figure 3
Figure 3 shows the test bench, which includes Arbin BT-5HC (Arbin, College Station, TX, USA) battery test equipment, a constant temperature and humidity chamber, tested Samsung ICR-18650-22P lithium-ion batteries (Samsung, Seoul, South Korea), and a host computer with MATLAB (R2016b, MathWorks, Natick, MA, USA) installed.The test chamber (Sanwood, Dongguan, China) controls the temperature and humidity environment of tested batteries.The battery test equipment has a voltage range of 5 V, four current ranges of 0.02 A, 0.5 A, 5 A, and 30 A, and measurement accuracy of current and voltage of ±0.02% full scale range.The battery test equipment controls the charge and discharge process and transmits the experimental data to the host computer in which the MATLAB is used to conduct the estimation process and analyze the experiment results.The tested batteries have a nominal capacity of 2150 mA, maximum continuous discharging

Figure 3 .
Figure 3. Structure of battery test bench.

Figure 4 .
The current, voltage, and SOC values are mainly collected in the experiment, and the data under NEDC driving cycle are used as the training data, and the data under the other two cycles are used for discussion sections.The batteries are discharged from 98% to 2% with the battery's surface temperature in the range from 25.42 °C to 26.48 °C, whose average value is 25.94 °C.

Figure 3 .
Figure 3. Structure of battery test bench.
and voltage, respectively.The DWT and IDWT processes are

Figure 7 .
Figure 7. SOC estimation with measurement noises under the NEDC.

Figure 7 .
Figure 7. SOC estimation with measurement noises under the NEDC.

Table 2 .
Identified parameters and tuning parameters of the comparative EKF estimation method.

Table 2 .
Identified parameters and tuning parameters of the comparative EKF estimation method.

Table 3 .
Mean and maximum SOC error with measurement noise under the NEDC.

Table 4 .
Mean and maximum SOC error of DWTLMWNN and EKF under NEDC, UDDS, and UKBC.

Table 4 .
Mean and maximum SOC error of DWTLMWNN and EKF under NEDC, UDDS, and UKBC.